Summary: We prove the existence of a positive solution for the three point boundary value problem on time scale given by
where is fixed and is singular at and possibly at , . We do so by applying a fixed point theorem due to J. A. Gatica, V. Oliker and P. Waltman [J. Differ. Equations 79, 62–78 (1989; Zbl 0685.34017)] for mappings that are decreasing with respect to a cone. We also prove the analogous existence results for the related dynamic equations , , and satisfying similar three point boundary conditions.